1. MECH300H Introduction to Finite Element Methods Lecture 10 Time-Dependent Problems

2. In general, Key question: How to choose approximate functions? Two approaches:

3. Model Problem I – Transient Heat Conduction Weak form:

4. Transient Heat Conduction let: and ODE!

5. Time Approximation – First Order ODE Forward difference approximation - explicit Backward difference approximation - implicit

6. Time Approximation – First Order ODE  - family formula: Equation

7. Time Approximation – First Order ODE Finite Element Approximation

8. Stability of – Family Approximation Stability  Example

9. FEA of Transient Heat Conduction  - family formula for vector:

10. Stability Requirment where Note: One must use the same discretization for solving the eigenvalue problem.

11. Transient Heat Conduction - Example

18. Model Problem II – Transverse Motion of Euler- Bernoulli Beam Weak form: Where:

19. Transverse Motion of Euler-Bernoulli Beam let: and

20. Transverse Motion of Euler-Bernoulli Beam

21. ODE Solver – Newmark’s Scheme where Stability requirement: where

22. ODE Solver – Newmark’s Scheme Constant-average acceleration method (stable) Linear acceleration method (conditional stable) Central difference method (conditional stable) Galerkin method (stable) Backward difference method (stable)

23. Fully Discretized Finite Element Equations

24. Transverse Motion of Euler-Bernoulli Beam